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\multicolumn{2}{c} {\bfseries Problem Set 9} \\
& \\
ECON 772001 - Math for Economists & Peter Ireland \\
Boston College, Department of Economics & Fall 2018 \\
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\multicolumn{2}{c} {Practice for the Midterm -- Not Collected or Graded}
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{\bfseries 1. Natural Resource Depletion}
Let $c_{t}$ denote society's consumption of an exhaustible natural resource at each date $t=0,1,2,...$, and suppose that a representative consumer gets utility from this resource as described by
\begin{equation}
\sum_{t=0}^{\infty} \beta^{t} \ln(c_{t}), \tag{1}
\end{equation}
where the discount factor satisfies $0 0
$$
on the terminal value of the stock of wealth.
Let $r_{t}$ be the interest rate earned on savings, or paid on debt, during each period $t=0,1...,T$; again as the notation suggests, this interest rate can vary over time. Then the consumer's stock of assets evolves according to
$$
k_{t+1} = k_{t} + w_{t} + r_{t}k_{t} - c_{t}
$$
during each period $t=0,1,...,T$. Allowing for the possibility of free disposal of wealth, which of course will never be optimal, these constraints can be written as
\begin{equation}
w_{t} + r_{t}k_{t} - c_{t} \geq k_{t+1} - k_{t} \tag{3}
\end{equation}
for all $t=0,1,...,T$.
So far, the set up of this problem generalizes the one that we studied in class by making labor income and the interest rate time-varying. Suppose, too, that the consumer's utility function also takes the more general, constant relative risk aversion form
\begin{equation}
\sum_{t=0}^{T} \beta^{t} \left( \frac{c_{t}^{1-\sigma}-1}{1-\sigma} \right), \tag{4}
\end{equation}
where $\sigma>0$ and $0